Endomorphism algebras of Jacobiansellenber/EndJacAjour.pdf · 2005. 7. 28. · ENDOMORPHISM...

Endomorphism algebras of Jacobians

Jordan S. Ellenberg

Department of Mathematics, Princeton UniversityFine Hall, Washington Road

Princeton, NJ 08544E-mail: [email protected]

Very little is known about the existence of curves, and families of curves,whose Jacobians are acted on by large rings of endomorphisms. In this paper,we show the existence of curves X with an injection

K ↪→ Hom(Jac(X), Jac(X))⊗Z Q,

where K is a subfield of even index at most 10 in a primitive cyclotomicfield Q(ζp), or a subfield of index 2 in Q(ζpq) or Q(ζpα ). This result generalizesprevious work of Brumer, Mestre, and Tautz-Top-Verberkmoes. Our curvesare constructed as branched covers of the projective line, and the endomor-phisms arise as quotients of double coset algebras of the Galois groups of thesecoverings. In certain cases, we show the existence of continuous families ofJacobians admitting the desired endomorphism algebras. At the end, we raisesome questions about upper bounds for endomorphism algebras of the typedescribed here, and ask about a “non-Galois version of Hurwitz’s theorem.”

2000 MSC classification: 14H40 (14H30)

Key Words: Jacobian, endomorphism, Hecke, double coset, Hurwitz.

INTRODUCTION

Let k be an algebraically closed field, A/k a principally polarized abelianvariety, and End0(A) the Q-algebra of endomorphisms of A. It has longbeen known which algebras E arise as End0(A) for some principally polar-ized abelian variety A [23, §21]. If we restrict our attention to Jacobiansof smooth curves of a given genus, however, the question is less well under-stood. Van der Geer and Oort remark [28, §5]:

“...one expects excess intersection of the Torelli locus and the loci corre-sponding to abelian varieties with very large endomorphism rings; that is,one expects that they intersect much more than their dimensions suggest.”

1

2 JORDAN S. ELLENBERG

This expectation has been borne out by many examples of families of curveswhose Jacobians are acted on by large algebras. Theorems of Tautz-Top-Verberkmoes [27], Mestre [19], and Brumer [2] produce, for each odd primep, families of curves X of genus (p−1)/2 such that End0(Jac(X)) containsthe real cyclotomic field Q(ζp + ζ−1

p ). Brumer also produces a family ofgenus 2 curves with Q(

√2) ⊂ End0(Jac(X)). Hashimoto-Murabayashi [15]

and Bending [1] have produced families of genus 2 curves whose Jacobiansare acted on by some quaternion algebras of small discriminant. De Jongand Noot give examples of infinitely many curves of genus 4 and 6 whose Ja-cobians have complex multiplication [4]. Ekedahl-Serre [9] produce manyexamples of curves of genus g with Q⊕g ⊂ End0(Jac(X)). Finally, Shi-mada [26] uses methods similar to ours to produce examples of curves oflarge genus whose Jacobians are acted on by quadratic fields.

In the present article we present a general procedure for constructingcurves whose Jacobians have large endomorphism algebras, and we showthat the endomorphisms produced in [27],[19], and [2], and most of thosein [9], arise via our construction. Furthermore, we show the existence ofsome new families of curves whose Jacobians are acted on by totally realfields.

Let F be a totally real number field. We say that a curve X over a field khas real multiplication by F if g(X) = [F : Q] and if X admits an injectionF ↪→ End0(Jac(X)). 1 We denote the index n subfield of Q(ζp) by Q(ζ(n)

p ).Our main theorem is the following.

Main Theorem. Let k be an algebraically closed field. Then

1. If p > 5 is a prime, and char k does not divide 2p, there exists a3-dimensional family of curves of genus (p− 1)/2 over k with real multipli-cation by Q(ζp + ζ−1

p ).2. If p is a prime congruent to 1 mod 4, and char k does not divide 2p,

then there exists a 1-dimensional family of curves of genus (p − 1)/4 overk with real multiplication by Q(ζ(4)


then there exists a 1-dimensional family of curves of genus (p − 1)/6 overk with real multiplication by Q(ζ(6)


then there exists a curve of genus (p− 1)/8 over k with real multiplicationby Q(ζ(8)


then there exists a curve of genus (p−1)/10 over k with real multiplicationby Q(ζ(10)

p ).1This notation is slightly non-standard in case char k divides the discriminant of F/Q;

see [8]. However, this case will not arise in the present work.

ENDOMORPHISM ALGEBRAS OF JACOBIANS 3

6. If p, q are distinct odd primes, and char k does not divide 2pq, thereexists a 1-dimensional family of curves of genus (p−1)(q−1)/2 over k withreal multiplication by Q(ζpq + ζ−1

pq ).

7. If p is an odd prime, and char k does not divide 2p, and α > 1 is aninteger, there exists a 2-dimensional family of curves of genus pα−1(p−1)/2over k with real multiplication by Q(ζpα + ζ−1

pα ).

Let X0/k be a smooth curve, and Y/X0 a Galois cover with Galois groupG. Let H be a subgroup of G and let X be the quotient curve Y/H. Thenthe Jacobian of X is acted on by the double coset algebra Q[H\G/H]. Wesay that the image of Q[H\G/H] in End0(X) is of Hecke type. The studyof endomorphism algebras of Hecke type is the main purpose of the presentpaper. In section 1, we explain how to compute the endomorphism algebraof Hecke type associated to a branched cover X → X0. We construct thefamilies of curves referred to above in sections 2 and 3, and show that thesefamilies have the stated dimensions in section 4.

The negative side of the question–that is, the question of which algebrascannot occur in End0(Jac(X))–is more mysterious. One can also ask weakerquestions about bounds on algebras which can occur as endomorphismalgebras of Hecke type attached to smooth curves. We discuss the currentstate of knowledge about such questions in section 5.

NOTATION

If V is a vector space acted on by a group H, denote by V H the subspaceof V fixed by H.

A “curve” will always be a nonsingular projective algebraic curve over afield unless specifically declared otherwise.

If A is an abelian variety, End0(A) is its Q-algebra of endomorphismsHom(A,A)⊗Z Q.

If p is a prime and n a divisor of p−1, we denote by Q(ζ(n)p ) the subfield

of index n in the cyclotomic field Q(ζp). For any integer m, we write Q(ζ+m)

for the real cyclotomic field Q(ζm + ζ−1m ).

If X → C is a dominant morphism of algebraic curves, the Galois groupof X/C is understood to mean the group of the Galois covering Y/C, whereY is the Galois closure of X/C.

1. ENDOMORPHISM ALGEBRAS OF HECKE TYPE

Let Y be a nonsingular projective algebraic curve over an algebraicallyclosed field k, and let G be a finite group which acts on Y . Write C for the


quotient of Y by G (Precisely, C is the nonsingular curve associated to theG-invariant subfield of k(Y ).)

The map

φ : G→ Aut(Y )

induces a map

e : Q[G]→ End0(Jac(Y )).

Now let H be a subgroup of G, and let X be the quotient curve Y/H.Define πH ∈ Q[G] to be

(1/|H|)∑h∈H

h.

We define the Hecke algebra Q[H\G/H] to be the subalgebra of Q[G]generated by πHgπH for all g ∈ G.

Remark 1. 1. We recall some basic facts about representations of finitegroups which we will need along the way. For proofs, see [18, 5.3]. Thegroup algebra Q[G] admits a direct product decomposition

Q[G] ∼=⊕i

Ai.

Each simple factor Ai is isomorphic to Mni(∆i), where ∆i is a centralsimple algebra whose center Ki is an abelian extension of Q. Let ejj be thematrix in Mni(∆i) which has a 1 in the (j, j) position and zeroes elsewhere.Then Vi := Mni(∆i)e11 is an irreducible representation of Q[G], and in factevery irreducible representation of Q[G] arises as Vi for exactly one i. Wecan think of Vi as a vector space over ∆op

i , in which case

Ai = End∆opi

(Vi).

In particular, πH commutes with the action of ∆opi on Vi; it follows that

V Hi = πHVi is a sub-∆opi -vector space of Vi, and that

πHAiπH = πH End∆opi

(Vi)πH = End∆opi

(V Hi ) ∼= Mdi(∆i)

for some di ≤ ni. So we have a decomposition

Q[H\G/H] ∼=⊕i

Mdi(∆i). (1.1)


The action e of Q[G] restricts to a natural action

Q[H\G/H]→ End0(e(πH) Jac(Y )). (1.2)

The natural map

Jac(X)→ Jac(Y )

restricts to an isogeny

Jac(X)→ e(πH) Jac(Y ).

Therefore, the action (1.2) yields an action

Q[H\G/H]→ End0(Jac(X)).

We refer to the image of Q[H\G/H] in End0(Jac(X)) as an endomor-phism algebra of Hecke type, and denote it by HX/C . In general, we say anelement α of End0(Jac(X)) is of Hecke type if there exists some branchedcover X → C such that α ∈ HX/C .

We note in passing that there exist Jacobians of smooth curves withendomorphisms which are not of Hecke type. The Jacobians of smoothgenus 3 curves form an open subscheme of the moduli space of principallypolarized abelian three-folds; in particular, there exist Jacobians of smoothgenus 3 curves which have real multiplication by non-abelian cubic fields.Among these, choose a curve X whose Jacobian has no endomorphismsother than those coming from the cubic field.

Let X → C be some surjective maps of curves, with Galois group G. NowHX/C is a quotient of Q[H\G/H], and as such is a direct sum of centralsimple algebras over abelian extensions of Q. In particular, HX/C cannotbe a non-abelian cubic extension of Q. So Jac(X) has endomorphismswhich are not of Hecke type.

Let ` be a a prime not equal to the characteristic of k. The natural map

End0(Jac(X))→ End0(H1(X,Q`))

is injective. Thus, to compute HX/C it suffices to study the representationof Q[H\G/H] on H1(X,Q`) ∼= πHH

1(Y,Q`). This representation, in turn,is determined by the action of G on H1(Y, Q`). Refer to this representationof G as ρY , and denote its character by χY . One can compute χY from thebranching data of the map Y → C, by means of Proposition 1.1 below.


Notation: for each g ∈ G, denote by χg the character of G induced fromthe trivial character on the cyclic group 〈g〉. Write χtriv for the trivialcharacter of G.

Remark 1. 2. Let U be the open curve C − {p1, . . . , pr}. If t ∈ k(C) isa uniformizer at pi, then we have a morphism

Spec k((t))→ U

which induces a map of etale fundamental groups

Gal(k((t))sep/k((t)))→ π1(U),

defined only up to conjugacy, since we have neglected to specify basepoints.The etale G-cover Y ×C U → U allows us to extend this diagram to

Gal(k((t))sep/k((t)))→ π1(U)→ G.

When we say “Y → C has monodromy gi at pi,” we mean that the compos-ite homomorphism above factors through the tame quotient of the Galoisgroup Gal(k((t))sep/k((t))), and that the image of a generator of tame iner-tia lies in the conjugacy class of gi. In case k = C, this definition coincideswith the natural topological one.

Proposition 1.1. Suppose the map Y → C is branched at r pointsp1, . . . pr ∈ C(k), with monodromy g1, . . . , gr, and suppose that each gi hasorder prime to char k. Then

χY = 2χtriv + 2(g(C)− 1)χ1 +∑i

(χ1 − χgi). (1.3)

Remark 1. 3. Note that χY depends only on the unordered set ofconjugacy classes of g1, . . . , gr. Suppose C = P

1. Then the existence ofa cover Y → C with monodromy conjugate to {g1, . . . , gr} implies thatthere are elements g′1, . . . g

′r and a permutation σ such that g′i is conjugate

to gσ(i) and g′1 . . . g′r = 1. If char k is prime to |G|, the converse is true; see

Proposition 1.2 below.

Remark 1. 4. It seems natural to call a character χ of a finite groupa Hurwitz character if χ is of the form (1.3) for some set of generators


g1, . . . , gr with g1 . . . gr = 1. It follows from Proposition 1.1 and the Rie-mann Existence theorem that a Hurwitz character is the character of arational representation (not merely a virtual rational representation) of G;a purely group-theoretic proof of this fact is given by Scott in [25]. We callsuch a representation a Hurwitz representation of G.

The question of which characters of a finite group are Hurwitz charactersis purely combinatorial, and seems quite mysterious; we will discuss thisquestion further in section 5.

We should also note that χY , while strictly speaking an `-adic character,takes values inQ; we will freely refer to the values of χY as rational numbersin what follows.

Proof. The proposition follows from the Lefschetz fixed point formulain etale cohomology; see for instance [21, V, Cor. 2.8]. The formula giventhere for χY is

χY = 2χtriv + 2(g(C)− 1)χ1 +∑i

api ,

where api is the Artin character associated to the branch point pi ∈ C(k).It thus suffices to show that

api = χ1 − χgi .

Let y1, . . . , ym be the points of Y (k) lying over pi. Take g ∈ G with g 6= 1.Then −(χ1(g)− χgi(g)) is the number of the yj which are fixed by g. Onthe other hand,

api(g) = −∑j

ij(g)

where ij(g) is the multiplicity of the fixed point of g at yj . Since char kis prime to the order of gi, the map Y → C is tamely ramified at y,so ij(g) = 1 for each yj which is a fixed point. ([21, p. 188].) Wehave thus shown that api(g) = χ1(g) − χgi(g) for all g 6= 1; since api

and χ1−χgi are both orthogonal to the trivial character, they are equal.

We recall for future reference Grothendieck’s computation of the prime-to-p fundamental group of a curve, which we use here as a replacement forthe Riemann existence theorem in characteristic p.

Proposition 1.2. Let k be an algebraically closed field. Let G be afinite group whose order is prime to char k, or an arbitrary finite groupif char k = 0. If g1, . . . , gr are generators for G satsifying g1 . . . gr = 1,


there exists an Galois covering Y → P1 with Galois group G, branched at

r points with monodromy g1, . . . , gr.

Proof. Immediate from [11, XIII,2.12].

We will give examples in the following sections to show that judiciouschoices of {G,H, g1, . . . , gr} yield interesting endomorphism algebras ofHecke type.

2. METACYCLIC GROUPS

Let F be a totally real number field. In this section, we use coversof P1 with metacyclic Galois groups in order to produce curves with realmultiplication by certain totally real number fields.

Let p be a prime, and n be an integer such that n|p − 1. Let G = Gp,nbe the metacyclic group

〈σ, α : σp = αn = 1, ασα−1 = σk〉

where k is an element of order n in (Z/pZ)∗. Let H be the subgroup of Ggenerated by α. Let ` be a prime not equal to char k. For consistency withetale cohomology, we will take Q` as the algebraically closed characteristic0 base field in which we compute the irreducible representations of G.

The irreducible Q`-representations of G fall into two types:

• n one-dimensional representations Va, indexed (non-canonically) bya ∈ Z/nZ; these arise as compositions

G→ Z/nZ→ Q∗`

where the second map sends 1 to ωan, for some fixed nth root of unityωn ∈ Q`.• (p− 1)/n representations of dimension n, which are induced from 〈σ〉.

We denote by W the direct sum of all the n-dimensional irreducibleQ`-representations of G.

Let g1, . . . , gr be non-trivial elements of G, and for each i in 1, . . . , r letdi be either 0 (if gi has order p) or n/ord(gi) (if gi has order dividing n.)Note that di determines the cyclic group generated by gi up to conjugacy.Recall that, for any g ∈ G, we denote by χg the character of G inducedfrom the trivial character on 〈g〉. We can compute

χgi = diχ(W ) +∑

a∈Z/nZ:adi=0

χ(Va).


Suppose now that k is an algebraically closed field of characteristic primeto |G|. Let Y be a Galois cover of P1, with Galois group G, branched at rpoints with monodromy g1, . . . , gr. It follows from Proposition 1.1 that

χY = mχ(W ) +∑

a∈Z/nZ

caχ(Va)

where

ca ={−2 + |{i : adi 6= 0}| a 6= 00 a = 0

and

m = −2n+∑i

(n− di).

Let X be the quotient curve Y/H. Since no non-trivial element of Va isfixed by H for a 6= 0, we have

H1(X, Q`) ∼= (WH)⊕m

as Q`[H\G/H]-modules.Let WQ be the (unique) Q-representation of G such that WQ⊗Q Q` = W .

Then the map

Q[H\G/H]→ End(H1(X, Q`)H) = End((WH)⊕m)

factors through

p : Q[H\G/H]→ End(WHQ

).

An element of Q[H\G/H] acts as the zero endomorphism of Jac(X) if andonly if it acts as zero on H1(X, Q`)H , which is to say if and only if it is thekernel of p. We conclude that HX/P1 is isomorphic to the image of p. Nowthe image of Q[G] in End(WQ) is isomorphic to the n × n matrix algebraover Q(ζ(n)

p ), and πH maps to a projection of rank 1. It follows that theimage of Q[H\G/H] = Q[πHGπH ] in End(WH

Q) is isomorphic to Q(ζ(n)

p ).Note that the dimension of WH is (p− 1)/n. So if n is even and m = 2,

we have exhibited a real field of degree (p− 1)/n acting on Jac(X), whichis of dimension (p− 1)/n. So X has real multiplication.

There are only finitely many possibilities for n and d1, . . . , dr satisfyingm = 2. Of these, only nine possibilities can arise from Gp,n by means ofthe above construction:


1a. n = 2; {di} = {0, 0, 1, 1}1b. n = 2; {di} = {0, 1, 1, 1, 1}1c. n = 2; {di} = {1, 1, 1, 1, 1, 1}2a. n = 4; {di} = {0, 1, 1}2b. n = 4; {di} = {1, 1, 2, 2}3a. n = 6; {di} = {2, 2, 3, 3}3b. n = 6; {di} = {1, 1, 2}4. n = 8; {di} = {1, 1, 4}5. n = 10; {di} = {1, 2, 5}.

The first three cases give examples of curves whose Jacobians have realmultiplication by real cyclotomic fields Q(ζ+

p ). The families of curves de-scribed by Tautz, Top, and Verberkmoes [27] fall under cases 1a and 1babove. In case 1c, the curve Y is a cover of P1 whose Galois group is thedihedral group D2p. The subcover corresponding to Z/pZ ⊂ D2p is a dou-ble cover of P1 branched over six points–that is, a genus 2 curve C ′. Thecurves X in case 1c are exactly those produced by Brumer [2] as quotientsof etale covers of genus 2 curves. The curves produced by Mestre in [19] arespecial cases of case 1c, arising when C ′ admits a non-constant morphismto an elliptic curve.

In cases 2b and 3a, the argument above results in the following twopropositions.

Proposition 2.1. Let g2, g′2, g4, g

′4 be elements of Gp,4, of orders 2, 2, 4, 4

respectively, satisfying g2g′2g4g

′4 = 1. Let H be an order-4 subgroup of Gp,4.

Let k be an algebraically closed field of characteristic greater than 2.Suppose Y/k is a Galois cover of P1/k, with Galois group Gp,4, branched

at four points with monodromy g2, g′2, g4, g

′4, and let X = Y/H.

Then X has real multiplication by the index-4 subfield of Q(ζp).

Proposition 2.2. Let g2, g′2, g3, g

′3 be elements of Gp,6, of orders 2, 2, 3, 3

respectively, satisfying g2g′2g3g

′3 = 1. Let H be an order-6 subgroup of Gp,6.

Let k be an algebraically closed field of characteristic greater than 3.Suppose Y/k is a Galois cover of P1/k, with Galois group Gp,6, branched

at four points with monodromy g2, g′2, g3, g

′3, and let X = Y/H.


Case 2a provides further examples of a curve with real multiplication byQ(ζ(4)

p ); likewise, case 3b provides further examples of real multiplicationby Q(ζ(6)

p ). Cases 4 and 5 bring new fields of real multiplication into thepicture; we record the results in the following propositions.


Proposition 2.3. Let g1, g2, g3 be elements of Gp,8, of orders 2, 8, 8respectively, satisfying g1g2g3 = 1. Let H be an order-8 subgroup of Gp,8.Let k be an algebraically closed field of characteristic greater than 2.

Suppose Y/k is a Galois cover of P1/k, with Galois group Gp,8, branchedat three points with monodromy g1, g2, g3, and let X = Y/H.


Proposition 2.4. Let g1, g2, g3 be elements of Gp,10, of orders 2, 5, 10respectively, satisfying g1g2g3 = 1. Let H be an order-10 subgroup of Gp,10.Let k be an algebraically closed field of characteristic prime to 10.

Suppose Y/k is a Galois cover of P1/k, with Galois group Gp,10, branchedat three points with monodromy g1, g2, g3, and let X = Y/H.


We note in the following Proposition that the covers described aboveactually do exist.

Proposition 2.5. Suppose char k does not divide 2p (resp. 6p, 2p, 10p).Then there exists a cover Y of P1/k with the branching properties describedin Proposition 2.1 (resp.2.2, 2.3, 2.4).

Proof. Immediate from Proposition 1.2.

Note that we have now completed the proof of cases 4 and 5 of the MainTheorem.

3. MORE DIHEDRAL GROUPS

In this section, we produce some curves whose Jacobians have real mul-tiplication using dihedral groups of order 2m, where m is odd but notnecessarily prime. Let G = Dm be the dihedral group of order 2m:

〈σ, τ : σm = τ2 = 1, τστ−1 = σ−1〉

Let H be the subgroup generated by τ .The group G has two irreducible Q`-representations of dimension 1, and

(m − 1)/2 of dimension 2. Denote by V1 and V−1 the representations ofdimension 1, and by W the direct sum of the 2-dimensional representations.Note that W splits over Q into a direct sum of representations

W =⊕

d|m:d6=1

Wd


where dimWd = φ(d), and Wd is the base change to Q` of an irreducibleQ-representation Wd;Q. Now the image of Q[G] on End(Wd;Q) is a 2 × 2matrix algebra over Q(ζ+

d ), and πH is sent to a rank 1 projection. So theimage of Q[H\G/H] = Q[πHGπH ] in End(Wd;Q) is isomorphic to Q(ζ+

d ).

Suppose m = pq, where p and q are distinct odd primes. Let g1, g2, g3, g4

be elements of G conjugate to τ, τ, σp, σq respectively, and satisfying

g1g2g3g4 = 1.

(For instance, we can take g1 = τ, g2 = τσ−q−p, g3 = σp, g4 = σq.) Let kbe an algebraically closed field of characteristic prime to |G|, and let Y bea Galois cover of P1, branched at 4 points with monodromy g1, g2, g3, g4.It then follows from Proposition 1.1 that

(ρY )H = (WHpq )⊕2.

Let X be the quotient curve Y/H. Then X is a curve of genus (1/2)(p −1)(q − 1). By the same argument as the one preceding Proposition 2.1,the image of Q[H\G/H] in End(Jac(X)) is isomorphic to its image inEnd(Wpq;Q), which is Q(ζ+

pq). So X has real multiplication by Q(ζ+pq).

Now consider the case m = pα, with p an odd prime and α > 1. Letg1, g2, g3, g4, g5 be elements of G conjugate to τ, τ, τ, τ, σp

α−1respectively,

and satisfying

g1g2g3g4g5 = 1.

For instance, we may take g1 = g2 = g3 = τ, g4 = τσ−pα−1

, g5 = σpα−1

. Letk be an algebraically closed field of characteristic prime to |G|, and let Y bea Galois cover of P1, branched at 5 points with monodromy g1, g2, g3, g4, g5.

Another application of Proposition 1.1 gives

(ρY )H = (WHpα)⊕2.

Again setting X = Y/H, we have X a curve of genus (1/2)(pα − pα−1)whose Jacobian has real multiplication by Q(ζ+

pα).One can check that the dihedral groups of orders 2p, 2pq, and 2pα are

the only ones which produce curves with real multiplication via the abovestrategy.

4. FAMILIES OF JACOBIANS WITH REALMULTIPLICATION


In the sections above, we have exhibited curves with real multiplica-tion by various totally real abelian number fields. These curves arose asbranched covers X → P

1 with specified ramification above the branchpoints. By moving the branch points, we naturally expect to obtain con-tinuously varying families of curves X with real multiplication.

The problem is that, a priori, we do not know that moving the branchpoints alters the curve. For instance, it might be the case that there is asingle curve X such that every 4-branched cover of P1 with Galois groupGp,4 and monodromy elements of order 2, 2, 4, 4 is isomorphic to X! Ourgoal in this section is to prove that such annoying circumstances do not, infact, arise.

For the length of this section, we fix:

• a finite group G with a subgroup H;• an algebraically closed field K with characteristic prime to |G|;• an r-branched Galois cover Y/K → P

1/K with Galois group G andmonodromy elements g1, . . . , gr.

Let χY the character of G acting on H1(Y, Q`), as computed in Propo-sition 1.1.

We let X = Y/H be the quotient curve, and π : X → P1 the projection

map. In X×X we have a divisor D = X×P1X; concretely, the closed pointsof D are the points (x1, x2) : π(x1) = π(x2). The irreducible componentsof D are in bijection with the double cosets HgH of H in G. Write Dg

for the component of D corresponding to HgH. An important step in theargument below is the observation that, in most of the cases we consider, Dg

has negative self-intersection. The self-intersection is computed by meansof the following proposition.

Proposition 4.1. The self-intersection of Dg is

|H|−2∑

g1,g2∈HgH2− χY (g1g

−12 ).

Remark 4. 1. The summand 2 − χY (g1g−12 ) can be thought of as the

number of fixed points of the action of g1g−12 on Y , whenever g1 6= g2.

Proof. Let Dg be the graph of g in Y ×Y . It follows from the Lefschetzfixed point theorem in etale cohomology [21, VI,12.3] that

Dg1 · Dg2 = Dg1g−12· D1 =

2∑i=0

(−1)iTr(g1g−12 |Hi(Y, Q`)) = 2− χY (g1g

−12 ).


Moreover, if f : Y → X is the quotient map, we have

(f × f)∗Dg =⋃

g1∈HgHDg1 .

The result now follows.

Suppose now that K has an algebraically closed subfield k, such thatK has finite transcendence degree over k; we may think of k as a field ofconstants and K as the algebraic closure of a function qyfield over k. Inparticular, we will say that the branched cover π : X/K → P

1K is non-

varying if it fits into a diagram

X X −−−−→ X0

π

y y yπ0

P1K

α−−−−→ P1K

g−−−−→ P1k

where the right-hand square is Cartesian, g is the base change of the struc-ture map SpecK → Spec k, and α is some automorphism of P1

K .We show below that, under a certain numerical condition on the rami-

fication of π, that “a branched cover from a curve defined over k to P1 isnon-varying.”

Theorem 4.1. Suppose X = X0 ×k K for some curve X0/k. Supposefurthermore that there exists a double coset HgH with cardinality |H|2 suchthat ∑

g1,g2∈HgH2− χY (g1g

−12 ) < 0. (4.4)

Then π : X → P 1K is non-varying.

Proof. The basic algebro-geometric ingredient of the proof is the factthat divisors with negative self-intersection cannot vary continuously. Moreprecisely, we need the following lemma.

Lemma 4.1. Let S/k be a smooth projective algebraic surface over analgebraically closed field. Let K/k be a algebraically closed extension of kof finite transcendence degree. Suppose that C ↪→ SK = S ×k K is anirreducible closed curve with negative self-intersection. Then C = C0×kKfor some irreducible closed curve C0 ↪→ S.


Proof. Let U be a variety over k, the algebraic closure of whose functionfield is K. Let CU be a closed subscheme of S×kU whose geometric genericfiber is C. Replacing U by an open subscheme of U , we may assume thatU is affine and that the fibers of CU → U are irreducible curves.

Now let P be a k-point of U . Write C0 for the restriction of C to P ;define C ′ = C0×P K and C ′U = C0×P U . Note that C ′, being smooth andconnected, is an irreducible curve in SK . We claim that C = C ′.

To prove the claim, observe that the intersection product can be ex-pressed in terms of Euler characteristics [22, Lecture 12]:

(C0 · C0) = χ(OS)− χ(OS(−C0))− χ(OS(−C0)) + χ(OS(−2C0))

and

(C · C ′) = χ(OSK )− χ(OSK (−C))− χ(OSK (−C ′)) + χ(OS(−C − C ′)).

Recalling that S is projective, we may think of OS ,OS(−CU ),OS(−C ′U ),and OS(−CU −C ′U ) as coherent sheaves on PNU . Now the fact that Hilbertpolynomials of coherent sheaves are locally constant in flat families impliesthat

(C · C ′) = (C0 · C0) = (C · C) < 0.

So CK and C ′K must share an irreducible component; since both divisors areirreducible curves, they must coincide.

We first consider the case in which H is trivial–that is, X = Y is aGalois cover of P1

K . Note that in this case the numerical condition (4.4)says precisely that g(X) ≥ 2.

Let C be the divisor X×P1X ↪→ X×KX. Then every component of C isa translate of the diagonal by an automorphism of X×KX; since g(X) ≥ 2,every component of C has negative self-intersection. From Lemma 4.1 wehave that C is the base change of a divisor C0 ⊂ X0 ×X0.

Now the intersection of C0 with a fiber x × X0 is a set Sx of n = |G|points in X0 (counted with multiplicity). One can thus define a morphism

η0 : X0 → SymnX0

which sends x to Sx. Let F0 be the image of η0, and write F, η for F0 ×kK, η0 ×k K. Now η(x) = η(y) if and only if π(x) = π(y), which is to saythat η factors as

Xπ→ P

1 α→ F,


where α is an isomorphism. Then the diagram

X X −−−−→ X0

π

y η

y yη0

P1K

α−−−−→ Fg−−−−→ F0

yields the desired result.

We now return to the general case. Let fg : Dg → Dg be the restrictionto Dg of the map f × f : Y × Y → X ×X. We claim that fg is genericallyone-to-one. It suffices to consider closed points. Suppose

(f(y), f(g(y))) = (f(y′), f(g(y′))).

Then y′ = h1(y) and g(y′) = h2(g(y)) for some h1, h2 ∈ H. So g(h1(y)) =h2(g(y)).

Suppose y (whence also y′) is not fixed by any nontrivial element ofG. Then h−1

2 gh1 = g. Since HgH has cardinality |H|2, this implies thath1 = h2 = 1, so y′ = y. We have shown that fg is one-to-one on closedpoints away from a closed subset of Dg; it follows that fg is genericallyone-to-one, as claimed. In particular, since Dg is isomorphic to the smoothcurve Y , we have that Y is a nonsingular model for the curve Dg.

It follows from Proposition 4.1 that Dg has negative self-intersection.Applying Lemma 4.1 with S = X0 ×X0 and C = Dg, we find that Dg =D0 ×k K for some curve D0 ∈ X0 ×k X0. Since Y is the normalization ofDg, we also have Y = Y0 × k. Write p = π ◦ f : Y → P

1K . Then p is a

Galois cover. Observe that 2 − χY (g) is a rational number greater than2−2g(Y ) for all g ∈ G; so (4.4) implies that g(Y ) > 2. It now follows fromthe H = {1} case treated above that p is non-varying. After composing pwith an automorphism of P1

K , we may therefore assume that p is the basechange of a Galois cover p0 : Y0 → P

1k. This means that the action of G on

Y is the base change of an action of G on Y0. Thus, p0 factors through amap π0 : Y0/H → P

1k, and π is evidently isomorphic to the base change of

π0. This yields the desired result.

We will now prove that the theorems of the previous sections actuallyproduce infinite families of curves with real multiplication. Our main toolwill be Theorem 4.1; however, in the cases where the field of real multipli-cation is Q(ζ(4)

p ) or Q(ζ(6)p ), we instead use a degeneration argument which

yields a slightly stronger result than the application of Theorem 4.1 would.Suppose K/k is an extension of algebraically closed fields of transcen-

dence degree n. For each g ≥ 2, let Mg be the moduli space of smooth


genus g curves. We say a genus g curve X/K is an n-dimensional familyof curves over k if the map SpecK ↪→ Mg(K) induced by X does notfactor through any SpecL, where L is an algebraically closed subextensionof K/k of transcendence degree less than n over k.

Corollary 4.1. Let k be an algebraically closed field. Then

1.If p > 5 is a prime, and char k does not divide 2p, there exists a 3-dimensional family of curves with real multiplication by Q(ζ+

p ).2.If p is a prime congruent to 1 mod 4, and char k does not divide 2p,

there exists a 1-dimensional family of curves with real multiplication byQ(ζ(4)

p ).3.If p is a prime congruent to 1 mod 6, and char k does not divide 6p,

there exists a 1-dimensional family of curves with real multiplication byQ(ζ(6)

p ).6.If p, q are distinct odd primes, and char k does not divide 2pq, there

exists a 1-dimensional family of curves with real multiplication by Q(ζ+pq).

7.If p is an odd prime, and char k does not divide 2p, and α > 1 is an in-teger, there exists a 2-dimensional family of curves with real multiplicationby Q(ζ+

pα).

Proof. We will begin by treating cases 1, 6, and 7 above, using Theo-rem 4.1. Cases 2 and 3 require us to make an argument on the boundaryof the moduli space of curves, which we carry out in the second section ofthe proof.

Let Kr be the algebraic closure of the function field of the moduli spaceM0,r/k of genus 0 curves with r marked points, where r ≥ 4. Then Kr

has transcendence degree r − 3 over k. Moreover, the inclusion

ι : SpecKr →M0,r

gives us a set of r points p1, . . . , pr on P1Kr

. Suppose π : X → P1Kr

is a coverbranched at p1, . . . , pr. Let p : Y → P

1Kr

be the Galois closure of π. Asusual, let G be the Galois group of p and H the subgroup correspondingto X via the Galois correspondence. Suppose that G,H, χY satisfy thenumerical condition (4.4) of Theorem 4.1. Then X is an (r−3)-dimensionalfamily of curves over k. For suppose otherwise; then X is isomorphic toX0×LKr, where L/k is an algebraically closed subextension of Kr/k withtranscendence degree less than r − 3. It then follows from Theorem 4.1that π is non-varying, which is to say that it is isomorphic the base changeto Kr of a cover π0 : X0 → P

1L. This means, in turn, that the branch locus


of π (i.e. the set of points {p0, . . . , pr}) is the base change to Kr of thebranch locus of π0. That is, the map

ι : SpecKr →M0,r

factors through SpecL, which contradicts the hypothesis that ι is a geo-metric generic point of M0,r.

It follows that, in each case of the corollary, we only need to demonstratethat there exists a branched cover π : X → P

1Kr

branched at p1, . . . , prwhich has the desired real multiplication and which satisfies (4.4).

We will now work case by case. Let F be the field of real multiplication.Case 1: F = Q(ζ+

p ). We use the branched cover described in section 2with real multiplication by F . (Recall that this is the cover described byBrumer [2].) Here r = 6, the Galois group G is the dihedral group Gp,2,the subgroup H is generated by an involution τ , and the monodromy ateach branch point is conjugate to τ . It follows from Proposition 1.1 that

χY = 2χtriv − 2χ1 + 6(χ1 − χτ ) = 2χtriv + 4χ1 − 6χτ .

So

• χY (1) = 2p+ 2;• χY (τ) = −4;• χY (σa) = 2 for all a prime to p.

The double coset HσH has cardinality 4. As g1, g2 run through HσH, thequotient g1g

−12 is trivial in 4 cases, conjugate to τ in 8 cases, and conjugate

to a nontrivial power of σ in 4 cases. So we have∑g1,g2∈HσH

2− χY (g1g−12 ) = 4(−2p) + 8 · 6 = 48− 8p,

which is negative once p > 5. Note that the assertion of Corollary 4.1 isin fact false for p = 5, since the moduli space of abelian surfaces with realmultiplication by Q(

√5) is only 2-dimensional.

Case 4: F = Q(ζ+pq). Take r = 4 and let X be the branched cover

described in Section 3 with real multiplication by F . In this case, Propo-sition 1.1 shows that

χY = 2χtriv − 2χ1 + 2(χ1 − χτ ) + (χ1 − χσp) + (χ1 − χσq ).

So

• χY (1) = 2(p− 1)(q − 1)


• χY (τ) = 0;• χY (σa) = 2 for all a prime to pq.

The sum in (4.4) is therefore

4(2− 2(p− 1)(q − 1)) + 8 · 2 = 24− 8(p− 1)(q − 1),

which is negative for all odd primes p, q.Case 5: F = Q(ζ+

pα). Take r = 5 and let X be the branched coverdescribed in Section 3 with real multiplication by F . Here, Proposition 1.1tells us that

χY = 2χtriv − 2χ1 + 4(χ1 − χτ ) + χ1 − χσpα−1 .

So

• χY (1) = 2 + 2pα − 2pα−1

• χY (τ) = −2;• χY (σa) = 2 for all a prime to p.

It follows that the sum in (4.4) is

4(2pα−1 − 2pα) + 8 · 4 = 32− 8(p− 1)pα−1,

which is negative for any odd prime p and any α > 1.

We now turn our attention to the cases F = Q(ζ(n)p ), where n = 4, 6.

The methods above will show the existence of 1-dimensional families ofcurves with real multiplication by F for all sufficiently large p congruent to1 mod n. In order to prove the desired result for all such p, we introducea different approach.

Note that to prove our family of curves is 1-dimensional, it suffices toprove it is non-constant. In turn, to prove the family is non-constant, it suf-fices to prove that it degenerates to a nodal curve (since the generic memberof the family is smooth.) In order to study the degeneration of our familyof branched covers, we need to recall some facts about compactifications ofHurwitz moduli spaces.

Let C/k be a connected stable r-pointed curve; that is, X has only nodalsingularities, and every rational component of the normalization of X hasat least three points lying over either marked or nodal points on X. Anadmissible cover of a stable r-pointed curve C is a finite morphism Y → Cwhich is etale over all smooth unmarked points of C, and which displays acertain local behavior over the nodes of C; the precise definition will notconcern us here. (See [14, 3.G] for more information.) If C is a smoothcurve, an admissible cover of C is just a smooth r-branched cover in theusual sense. The central fact we need is that the moduli stack of admissiblecovers is a compactification of the moduli stack of smooth branched covers.


Lemma 4.2. Let A be a strictly henselian discrete valuation ring withfraction field K. Let B → A be a stable r-pointed curve whose generic fiberBK is smooth, and let πK : XK → BK be an admissible cover. Supposethat the Galois closure of πK has Galois group G with order in A∗. Thenthere exists a tamely ramified extension A′ of A, with fraction field K ′,such that πK′ extends to an admissible cover π : XA′ → BA′ .

Proof. Argue as in [20, §3.13]. The only difference is that the hypothesis|G| ∈ A∗ replaces Mochizuki’s hypothesis d! ∈ A∗.

We use again the notations introduced in section 2. Let G be either Gp,4or Gp,6, let g be an element of order 4 or 3, respectively, and let F be thefield Q(ζ(4)

p ) or Q(ζ(6)p ) respectively. Let h be an element of G of order 2.

We think of G as embedded in the symmetric group Sp by means of theaction of G on G/H.

Let K be the algebraic closure of k(t). By Proposition 1.2, there existsa Galois cover Y → P

1K branched at the points 0, 1,∞, and t, with mon-

odromy conjugate to g, g−1, h−1, h respectively. Write X = Y/H. Thiscover has a model XU → P

1U , where U is an open subscheme of a finite

etale cover of Spec k[t], and XU → P1U is smooth. Note that X has real

multiplication by F , by the results of section 2.Let R be the integral closure of k[t] in the fraction field of U , and let

S = SpecR. First of all, by the properness of the moduli space M0,4

of 4-pointed stable genus 0 curves, there is a finite cover S′ of S and astable 4-pointed genus 0 curve B → S′ whose geometric generic fiber isP 1K marked at 0, 1,∞, and t. Let U ′ = S′ ×S U . Then XU ′ → BU ′ is

a a smooth 4-branched cover. It follows from Lemma 4.2 that, possiblyafter another finite extension of S′, the cover XU ′ can be extended to anadmissible cover XS′ → B. For simplicity of notation, replace S by S′.

Let s ∈ S be a closed point of S lying over the point t = 0 of Spec k[t].Then Bs is a nodal 4-pointed curve of genus 0; any such curve consists oftwo rational curves meeting at the node, with two marked points lying oneach of the irreducible components. Let Xs be the fiber of XS over s, sothat πs : Xs → Bs is an admissible cover. Let C,C ′ be the two irreduciblecomponents of Bs. Then π−1

s (C) → C is a branched degree p cover (notnecessarily connected) of a smooth genus 0 curve, which is etale away fromthe two marked points on C and the node C∩C ′. Call these points m0,mt,and n.

Now consider an infinitesimal neighborhood of s ∈ S, which is isomorphicto Spec k[[u]]. Since m0 is a smooth point of C, the completed local ring ofB at m0 is isomorphic to Spec k[[u, v]]. The marking divisor 0 is a sectionfrom S to B, which is locally given by the map of rings k[[u, v]] → k[[u]]


sending v to 0. Let SpecA be the scheme completing the Cartesian square

SpecA −−−−→ Xy ySpec k[[u, v]] −−−−→ B.

Then it follows from Abhyankar’s Lemma that

A ∼=d⊕i=1

k[[u, v]][V ]/(V ni − v)

where the n1, . . . , nd are the lengths of the orbits of the element g of Sp.Now, restricting SpecA to the fiber u = 0, we see that the monodromyaround m0 in the cover πC : π−1

s (C) → C is conjugate to g. Similarly,πC has monodromy conjugate to h around mt. So the monodromy of πCaround n is g′h′, where g′ is some conjugate of g−1 and h some conjugate ofh−1. It is now an easy Riemann-Hurwitz computation that every compo-nent of π−1

s (C) has genus 0 or 1, and that the same is true of π−1s (C ′). In

particular, the stable model of Xs is not a smooth curve of genus (p−1)/n.We conclude that the generically smooth family of curves X has a fiber Xs

whose stable model is non-smooth; therefore, X is the desired non-constantfamily of curves with real multiplication by F .

One would like to have explicit equations, as in [27],[19] and [2], for thefamilies of branched covers π : X → P

1Kr

discussed above. These familiesof branched covers are parametrized by certain Hurwitz spaces, which aresmooth schemes of dimension r − 3 over k.

To give a rationally parametrized family of branched covers would be togive a branched cover X → P

1U with the specified monodromy g1, . . . , gr,

where U is an open subscheme of a rational variety. Such a cover wouldproduce a map from U to a Hurwitz space. In case r = 4, the Hurwitzspace is a union of curves, whose genera can be computed by passing tocharacteristic 0 and using combinatorial methods. A computer calculationshows that in the cases F = Q(ζ(4)

13 ),Q(ζ(6)13 ), the relevant Hurwitz spaces

are already genus 1 curves. Our methods thus cannot produce rationallyparametrized families of curves whose Jacobians have real multiplicationby Q(ζ(4)

13 ) and Q(ζ(6)13 ), and indeed, we have no reason to expect that such

families exist.The situation is different in the case F = Q(ζ+

pq). In fact, one canrather easily construct a rationally parametrized family of curves with realmultiplication by F .


Proposition 4.2. Let p and q be odd primes, let k be an algebraicallyclosed field of characteristic prime to 2pq, and let X/k be the non-singularcurve with affine model

Tpq(y/2) = P (x)/2(x− 1)p(x− λ2)q

where Tpq is the degree pq Tchebyscheff polynomial satisfying

Tpq((z + z−1)/2) = (zpq + z−pq)/2

and P (x) is the degree p+ q polynomial such that

P (w2) = (w − 1)2p(w − λ)2q + (w + 1)2p(w + λ)2q.

Then X is a curve of genus (1/2)(p− 1)(q − 1) with real multiplication byQ(ζ+

pq).

Proof. Let Y be the curve with affine model

zpq =(w − 1)p(w − λ)q

(w + 1)p(w + λ)q.

One easily checks that the map Y → P1 sending (z, w) to w2 realizes Y as

a cover of the line with Galois group Dpq, branched over 0, 1,∞, and λ2

with ramification of order 2, q, 2, p respectively. The action of Dpq on Y isgenerated by

τ : (z, w)→ (z−1,−w)

and

σ : (z, w)→ (ζpqz, w).

Let H be the subgroup of Dpq generated by τ . Then Y/H is evidently iso-morphic to X; it now follows from the arguments of section 3 that X has thedesired real multiplication.

Remark 4. 2. By choosing λ in a field k0 which is not algebraicallyclosed, one can arrange for X to have a model over k0; however, the en-domorphisms of Jac(X) will typically be defined only over an extension ofk0. The problem of determining fields of definition and fields of modulifor the families of curves constructed here is part of the active program ofconstructing models of branched covers over non-algebraically closed fields.


The interested reader should consult the papers of Debes, Douai, and Em-salem [6],[7].

5. QUESTIONS

The main goal of this paper has been to exhibit smooth curves whoseJacobians had many endomorphisms of Hecke type. The negative version ofthis question–what endomorphism algebras cannot act on the Jacobian of asmooth curve?–seems substantially more difficult. In fact, we do not knowof a single example of an algebra E which embeds in the endomorphismalgebra of a principally polarized abelian variety but which can be provennot to embed in the endomorphism algebra of the Jacobian of a smoothcurve. In this section, we frame some questions in the direction of boundson endomorphisms of Hecke type.

5.1. A non-Galois version of Hurwitz’s theorem?Hurwitz showed [16] that a curve of genus g > 1 over a field k of charac-

teristic 0 has an automorphism group of size at most 84(g − 1). In otherwords, if Y/k → C/k is a Galois cover with Galois group G, the dimensionof Q[G] is bounded linearly in g(Y ).

Now let Y,C,G,H,X be defined as in section 1. Let HX/C be the endo-morphism algebra of Hecke type; recall that HX/C is a homomorphic imageof Q[H\G/H] and a subalgebra of End0(Jac(X)). We will always supposethat X/C is a tamely ramified cover, so that Proposition 1.1 is in effect.

Question A: Is there a universal constant γ such that

dimHX/C ≤ γg(X)

for all X,C?

Hurwitz’s theorem implies that the answer to question A is yes, withγ = 84, if we impose the additional condition that H is trivial.

Question A can be rephrased purely in terms of combinatorial grouptheory. First of all, if π : X → C is a branched cover, we may composeπ with a map ψ : C → P

1. The endomorphism algebra of Hecke typeassociated to the branched cover ψ ◦ π : X → P

1 is easily seen to containHX/C . So Question A does not change if we replace C by P1. WhenC = P

1, the action of G on H1(Y,Q`) is described by a certain Hurwitzcharacter (see Remark 1.4) and the dimension of HX/C can be determinedfrom that character.


To be precise: let χ be a Hurwitz character of a finite group G, and letH be a subgroup of G. We can decompose χ as

χ =∑ψ

aψ(χ)ψ

where ψ ranges over the irreducible characters of G. If ψ is a character ofG (trivial or not), write ψ(H) for the inner product of χ with the characterχH induced from the trivial character on H. Now define

A(G,H, χ) =∑

ψ:aψ(χ)>0

ψ(H)2.

Let Y → P1 be a branched cover with Galois group G giving rise to the

Hurwitz character χ on H1(Y,Q`), and let X = Y/H. Then A(G,H, χ) isprecisely the dimension of HX/P1 . Note also that χ(H) = 2g(X).

We arrive at the following equivalent formulation for Question A.Question A’: Is there a universal constant γ such that

A(G,H, χ) ≤ γχ(H)

for all G,H, χ?

Results of Aschbacher, Guralnick, Neubauer, and Thompson give partialresults in the direction of question A′ in case H is a maximal subgroupof G, the intersection of whose conjugates is trivial. Such subgroups areclassified by a theorem of Ashbacher and Scott; the results of the four aboveauthors show that, in all but one case of the Aschbacher-Scott classification,[G : H] is bounded above by a linear function in χ(H) [13, §1]. Plainly,A(G,H, χ) ≤ [G : H], so in these cases Question A′ is answered in theaffirmative. For example, if G is solvable and H is subject to the aboverestrictions, then γ can be taken to be 210 [24]. More results in this directionwill appear in [12].

5.2. Decomposable JacobiansIn [9], Ekedahl and Serre ask whether there are curves over C of every

genus, or even of arbitrarily large genus, whose Jacobians are isogenous toproducts of elliptic curves. Let X/k be a curve over an algebraically closedfield of characteristic 0, and let d(X) be the supremum of dim(A) over allsimple isogeny factors A of Jac(X). Then we can rephrase Ekedahl andSerre’s question to read: what conditions on the genus of X are imposedby the condition d(X) = 1? More generally, we might ask:


Question B: Is there an increasing, unbounded function f : N → N

such that, for all smooth genus g curves X/k,

d(X) ≥ f(g)?

By taking C to be a genus g0 curve and X to be the maximal elementary2-abelian etale cover of C, one finds that such an f would have to be oforder at most c log g. Serre has proved that, if k is replaced by a finitefield F, and if d(X) is the maximal dimension of an F-simple isogeny factorof Jac(X) over F, then the answer to question B is yes. In fact, de Jonghas shown [5] that f(g) can be chosen on the order of (log log g)1/2 in thiscase. On the other hand, if k is taken to be an algebraically closed fieldof characteristic p, the existence of supersingular curves of arbitrarily largegenus shows that the answer to question B is no.

In some cases, we can use endomorphisms of Hecke type to bound d(X).Let X,Y,G,H,C be defined as in section 1, with |G| prime to char k.

Denote by M the Q[H\G/H]-module H1(X,Q`). Let V1, . . . , Vm be theirreducible Q-representations of G. Recall the decomposition

Q[H\G/H] ∼=∑i

Mdi(∆i). (5.5)

given in (1.1).For each i, write πi,j for the element

ejj ∈Mdi(∆i) ⊂ Q[H\G/H].

Then πi,j is an idempotent, and we can write

M ∼=∑i,j

πi,jM.

We call the subspace πi,jM a factor of M . Note that e(πi,j) Jac(X) is anabelian subvariety of Jac(X) of dimension (1/2) dimQ`(πi,jM).

Now define

d(X/C) = maxi,j

(1/2) dim(πi,jM).

(Note that the maximization over j is really redundant, since πi,jM andπi,j′M have the same dimension.)

The decomposition of Jac(X), up to isogeny, as

Jac(X) ∼⊕i,j

e(πi,j) Jac(X)


implies that d(X) ≤ d(X/C).A slight modification of d(X/C) is sometimes useful. We may suppose

that the first factor in the decomposition (5.5) is a copy of Q, which wemay think of as the quotient of Q[H\G/H] by its augmentation ideal. Sod1 = 1,∆1 = Q. Now define

d′(X/C) = maxi>1,j

(1/2) dim(πi,jM).

Note that π1,1 projects Jac(X) onto its G-invariant part, which is isogenousto Jac(C). So we have

Jac(X) ∼ Jac(C)⊕⊕i>1,j

e(πi,j) Jac(X),

which implies that

d(X) ≤ max(d(C), d′(X/C)).

Ekedahl and Serre give many examples of curves X with large genussuch that d(X) = 1. With the exception of some modular curves, theirexamples, like ours, are produced by analyzing the action of a finite groupon the Jacobian of a Galois cover of P1. In our notation, the Ekedahl-Serre examples are given as branched covers of low-genus curves C withd(C) = 1, and all have d′(X/C) = 1.

We are now motivated to ask whether this strategy has any chance ofproviding a negative answer to Question B.

Question C: Is there an increasing, unbounded function f : N→ N suchthat, for all smooth genus g curves X/k and all branched covers X → C,

d(X/C) ≥ f(g)?

We can rephrase question C as a question in combinatorial group the-ory. By composing the map X → C with an arbitrary map C → P

1, wemay think of X as a branched cover of the projective line; since HX/C iscontained in HX/P1 , we may restrict to the case C = P

1.Let G be a finite group, H a subgroup of G, and V be a Hurwitz repre-

sentation of G. We decompose V into irreducible Q-representations as

V =⊕i

V⊕ai(V )i .

Write χ for the character of V , and χi for the character of Vi.


Now V H is a Q[H\G/H]-algebra. Then

dimQ πi,jV H ={ai(V ) dimQ∆i = 〈χi, χ〉 di > 00 di = 0

Suppose Y/C is a cover of curves with Galois group G such that theaction of G on H1(Y,Q`) is given by V ⊗Q Q`. Let X = Y/H. Then thedimension of e(πi,j)(Jac(X)) is (1/2) dimQ πi,jV H .

This leads us to the following equivalent formulation of question C:

Question C’: Let G be a finite group, V a Hurwitz representation of G,and H a subgroup of G. Let Σ be the set of i in [1..m] such that di > 0.

Is there an increasing, unbounded function f : N → N, independent ofG,H, and V , such that

maxi∈Σ〈χi, χ〉 ≥ f(dim(V H))?

This problem seems rather difficult even for the case of Galois covers,where H is trivial.

We will discuss one special case, from which Ekedahl and Serre derivemany examples of curves with decomposable Jacobians([9, §4]). Let X/P1

be a Galois cover with group G, and suppose that G admits a non-trivialhomomorphism to Z/2Z, with kernel K. We thus obtain a diagram

X → X0 → P1,

where X0 is a hyperelliptic curve. Let V be the Hurwitz representation ofG corresponding to the cover X → P

1, and suppose that

maxi〈χi, χ〉 < B.

We will derive an upper bound, in terms of B, for g(X) = (1/2) dimV H .Let χ be the character of V , and χi the character of Vi. Now each

irreducible complex character ψ of G is a constituent of a unique χi. Inparticular,

maxψ〈ψ, χ〉 ≤ max

i〈χi, χ〉 < B.

Now if θ is an irreducible complex character of K, we have

〈θ, χ|K〉K = 〈IndGK θ, χ〉G.


Since IndGK θ is a sum of either one or two irreducible characters of G wehave that 〈θ, χ|K〉K < 2B.

Suppose furthermore that g(X0) ≥ 2 and that the cover X → X0 is etale,as is the case in Ekedahl and Serre’s examples. One has by Proposition 1.1that

χ|K = 2χtriv + 2(g(X0)− 1)χ1. (5.6)

(Here, χtriv and χ1 are the trivial and regular characters of K, not G.) Inparticular, if θ is a non-trivial irreducible complex character of K, then

〈θ, χ|K〉K = 2(g(X0)− 1)(θ(1)) < 2B. (5.7)

So K has no irreducible complex representations of dimension greaterthan B. Hence the irreducible complex representations of G have dimensionat most 2B. A theorem of Isaacs and Passman [17, (12.23)] now tells usthat G has an abelian subgroup A of index at most ((2B)!)2.

We can also bound the order of an element of G. Let g be an elementof G of some order n, and let W be an irreducible complex representationof G in which 〈g〉 acts faithfully. Since the dimension of W is at most2B, the field of definition of the character of W must have degree at leastφ(n)/2B over Q. In particular, if Vi is the irreducible Q-representation ofG containing W , then dimQ∆i is at least φ(n)/2B.

We know from the desription of χ|K in (5.6) that 〈χi|K,χ|K〉 > 0. Letτ be the character of G obtained by pullback from the non-trivial characteron Z/2Z. Denote χi ⊗ τ by χi′ . Then either χi or χi′ must occur withpositive multiplicity in the decomposition of χ; that is, either ai or ai′ ispositive. Note that ∆i′ = ∆i. We thus have

B > maxi〈χi, χ〉 = max

iai(V ) dimQ∆i > φ(n)/2B.

We conclude that φ(n) < 2B2, where n is the order of any element of G.This implies in turn that n < o(B2−ε).

Since K is the Galois group of a connected etale cover of X0, it canbe generated by 2g(X0) elements. Write A′ for A ∩ K. Then [K : A′] isat most ((2B)!)2, and it follows from the Reidemeister-Schreier theoremthat A′ can be generated by 2g(X0)((2B)!)2 + 1 − ((2B)!)2 elements. Forsimplicity, we say that A′ can be generated by 2g(X0)((2B)!)2 elements.Since every element of A′ has order at most o(B2−ε), we have

log |A′| = O(2g(X0)((2B)!)2 logB).

Furthermore, since g(X0) = O(B) by (5.7), we have

log |A′| = O(B logB((2B)!)2).


Since [G : A′] is bounded above by a constant times ((2B)!)2, we also havelog |G| = O(B logB((2B)!)2), whence

log log |G| = O(2 log(2B)!) = O(B logB).

Note that the genus of X is just 1 + (g(X0)− 1)|G|, which is O(B)|G|. So

log log g(X) = O(B logB),

from which we conclude that

d(X/P1) ≥ c log log g(X)log log log g(X)

for some absolute constant c.

ACKNOWLEDGMENTIt is my pleasure to thank Armand Brumer, who generously communicated his ideas to

me and thereby inspired this paper. I also thank Columbia University for the hospitalityI enjoyed while preparing much of the article. I thank Johan de Jong, Robert Guralnick,Martin Isaacs, Adam Logan, and Stefan Wewers for useful discussions, and the refereefor thorough and helpful comments on an earlier draft.

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